John D. Barrow grapples with infinity - a concept great minds have struggled to grasp - and charts the way it has shaped our view of the universe
The boundless, timeless and endless have attracted and confused human minds for thousands of years. From East to West, sophisticated cultures devised words to express the concept of infinity as well as arguments to include it or banish it from their models of the world. Aristotle distinguished clearly between "actual" and "potential" infinities. The actual variety, which involved boundless values of observable or measurable quantities here and now, were outlawed. On the other hand, he permitted the less threatening notion of a potentially infinite sequence, such as the unending list of positive whole numbers or an eternity of future time, where the infinity was neither achieved nor reached.
In many ways the attitude towards actual infinities mirrored that towards a vacuum. For Aristotle, the two were intimately connected because in an empty space there would be no resistance to motion and bodies would eventually move with infinite speed. Therefore, no perfect vacuum could exist in the physical universe. Medieval science devised ingenious arguments to avoid either. A "celestial agent" was imagined to act as a cosmic censor, ensuring that any opportunity to create a real vacuum or an infinity, even if permitted by the laws of mechanics, was never taken advantage of by nature.
Much has happened to change our conceptions of the infinite since those path-breaking arguments emerged. But it still challenges theologians, philosophers and scientists to understand it, cut it down to size, find out if it comes in different shapes and sizes, and decide whether we want to outlaw it or welcome it with open arms into our descriptions of the universe. Infinity is also very much a live issue. Physicists have spent the past 25 years in a quest for a so-called theory of everything that unites all the known laws of nature into a single mathematical statement. That search has been significantly guided by an attitude towards the existence of actual physical infinities.
In theories of particle physics, the appearance of an infinite answer to a question about the magnitude of a measurable quantity was always taken as a warning that you had made a wrong turn. For decades its inevitable appearance was managed by a strange subtraction procedure that removed the divergent part from the calculation to leave only a finite residue to compare with observations. Although the results of this so-called "renormalisation" process gave spectacularly good agreement with experiments, there was always deep unease that this ugliness could not be part of nature's economy. The true theory must be finite.
This all changed in 1984. Michael Green, now at Cambridge University, and John Schwarz, at the California Institute of Technology, showed that a particular type of physics theory could indeed be wholly finite. These "superstring" theories were based on the idea that the basic ingredients of the world were not "points" of mass or energy, but lines or loops called "strings". They were "super" because of a symmetry they possessed that allowed matter and radiation to be united. The upshot was that particles are just excited vibrations of the string and might one day have their exact masses and interaction properties precisely determined by calculating the energies of the natural vibrations of the superstring. The enthusiasm with which the new theories were embraced by physicists was a consequence of their ingenious banishment of infinities, a problem that had plagued their predecessors.
The path towards superstring theories awaits experimental endorsement. But the energy with which they have been pursued reflects the philosophy of scientists who believe that the appearance of an actual infinity in a physical theory is a signal that it is being stretched beyond its domain of applicability. The implicit solution is to upgrade the theory until the infinities are smoothed into large, but finite, quantities. Engineers, for example, know this well, exorcising the appearance of infinities in simple models of rapid aerodynamic flows by simply including more realism in the description of the air. The crack of a whip is caused by the sonic boom from the tip travelling faster than the speed of sound. A simple calculation that ignored the friction of air would say that this involved something changing infinitely quickly. But a more detailed modelling of the air flow properties turns this infinity into a very rapid but finite change.
Despite the general adoption of this infinities-mean-you-must-try-harder dictum in relation to physical theories, one area of science has been willing to take predictions of actual infinities more seriously. Cosmologists see a lot of infinities. Many are of the "potential" variety - the universe might be infinite in size, face an infinite future or contain an infinite number of stars. While they pose no local threat to the fabric of reality, we have to face up to Nietzsche's infinite replication paradox: if the universe is infinite and exhaustively random, then any event that has a finite probability of occurring here and now (such as you reading this article) must be occurring infinitely often elsewhere at this very moment. Moreover, for every history we have pursued here, all possible alternatives are acted out, wrong choices made simultaneously with right choices. This is a grave challenge to ethics and to the theology of almost every religion. Some find it so alarming that they regard it as a powerful argument for a finite universe. However, the finiteness of the speed of light insulates us from contact with our doubles. For all practical purposes we can only see and receive signals from a finite part of the universe.
The challenge to cosmologists does not end there, though. They also have to worry about "actual" infinities. For decades cosmologists have been happy to live with the notion that the universe of space and time began expanding from an initial big bang "singularity", where temperature, density and just about everything else, were infinite at some finite time in the past. Furthermore, when large stars exhaust their nuclear fuel and implode as a result of their own gravity, they appear doomed to reach a state of infinite density in finite time. But this is all neatly kept out of reach. Black holes are believed to be always shrouded by an "event horizon" - a surface of no-return through which things can fall in but not pass out - so that we can neither see the possible infinite density within nor feel its effects.
Roger Penrose, the Oxford University theoretical physicist, believes actual infinities do occur both at the start of the universe and at the centre of black holes. He once proposed that the laws of nature provide a form of cosmic censorship that ensures that such naked physical infinities are always enclosed by event horizons. This is reminiscent of the celestial agent invoked by medieval thinkers to avoid the creation of a perfect vacuum. By contrast, cosmologists with a particle-physics perspective tend to see these cosmological infinities merely as a signal that the theory has overextended itself and needs to be improved to exorcise these infinities.
As a result, we find much interest in the prospect of universes that bounce back into expansion if run backwards in time towards their apparent beginning. Our presently expanding universe is suspected of having arisen from the rebound, at finite density and temperature, of a previously contracting phase in its history.
From the outside, we cannot see what is happening inside a black hole. But if we fell in, we would be facing an uncertain fate as we approached the centre. Is there a real physical infinity, does energy slip away into another dimension of space, simply disappear into nothing or does it get soaked by exciting a never-ending sequence of vibrations of the superstrings at the core of all matter and energy? We just do not know. But again, the issue of finite versus infinite is a crucial guiding principle. Do we treat the appearance of an infinity as a signal to update our theory or do we treat it more seriously as an indication that new types of law govern infinite physical quantities, laws that could dictate how our universe began and how matter meets its end under the relentless implosion of gravity.
Cosmologists have another strange potential infinity to contemplate: the possibility of an infinite future. Is the universe on course to last forever? Its contemplation leads quickly to philosophy, for what does "forever" mean? And to biology and computer science - or can life, in any form, continue forever? And to the social sciences - and what would it mean socially, personally, mentally, legally, materially and psychologically for us to live forever? The last question, at least, is one to which we can all think of answers. In the long run, living forever might not prove as attractive as it seems at first. You might even welcome the televangelist who offers you the promise of finite life.
Mathematicians have also had to face up to the reality of infinity. Twice in the past 120 years, mathematics faced civil war over the matter, leaving many a casualty and much bitterness. Some wished to outlaw actual infinities and redefine boundaries to forbid all treatment of them as real "things". Journals were compromised and mathematicians ostracised.
The 19th-century German mathematician Georg Cantor first showed how to make sense of the paradoxes of infinity. He elegantly defined infinite collections as those that could be put in one-to-one correspondence with subsets of themselves. This enabled him to go on and answer deeper questions: can one infinity be bigger than another? Is there an ultimate infinity beyond which nothing bigger can be constructed or conceived, or do infinities go on forever? Cantor answered all these questions in precise ways but did not live long enough to see the fruits of his genius form part of the acknowledged body of mathematics. He was sidelined and undermined by influential finitist opponents, and for long periods he turned instead to the study of history and theology and suffered bouts of depression before his death in 1918.
Remarkably, theologians first seized on the importance of Cantor's work. They had long struggled to make sense of the infinities lurking within their doctrines. Is God alone infinite? Must he not be "bigger" than other more mundane infinities? Cantor's work revealed that there is a never-ending hierarchy of infinities, each unambiguously bigger than the last. He enabled us to distinguish between three different types of infinity: the mathematical, the physical and the transcendental. Some thinkers accept them all, some accept only some, some accept none.
The ancient philosophers, beginning with Zeno, were challenged by the paradoxes of infinities on many fronts. But what about philosophers today? What sort of problems do they worry about? There are live issues on the interface between science and philosophy that are concerned with whether it is possible to build an "infinity machine" that can perform an infinite number of tasks in a finite time. Of course, this simple question needs some clarification: what exactly is meant by "possible", "tasks", "number", "infinite", "finite" and, by no means least, by "time".
Classical physics appears to impose few physical limits on the functioning of infinity machines because there is no limit to the speed at which signals can travel or switches can move. Newton's laws allow an infinity machine. This can be seen by exploiting a discovery about Newtonian dynamics made in 1971 by the US mathematician Jeff Xia. First, take four particles of equal mass and arrange them in two binary pairs orbiting with equal but oppositely directed spins in two separate parallel planes. Now, introduce a fifth much lighter particle that oscillates back and forth along a perpendicular line joining the mass centres of the two binary pairs. Xia showed to everyone's surprise that such a system of five particles will expand to infinite size in a finite time.
How does this happen? The little oscillating particle runs back and forth between the binary pairs, each time creating an unstable meeting of three bodies. The lighter particle then gets kicked back and the binary pair recoils outwards to conserve momentum. The lighter particle then travels across to the other binary and the same ménage á trois is repeated there. This continues without end, accelerating the binary pairs apart so strongly that they become infinitely separated while the lighter particle undergoes an infinite number of oscillations in the process.
Unfortunately (or perhaps fortunately), this behaviour is not possible when relativity is taken into account. No information can be transmitted faster than the speed of light and gravitational forces cannot become arbitrarily strong in Einstein's theory of motion and gravitation; nor can masses get arbitrarily close to each other and recoil - there is a limit to how close separation can get, after which an "event horizon" surface encloses the particles to form a black hole. Their fate is then sealed - no such infinity machine could send information to the outside world.
But this does not mean that all relativistic infinity machines are forbidden. Indeed, the Einsteinian relativity of time that is a requirement of all observers, no matter what their motion, opens up some interesting new possibilities for completing infinite tasks in finite time. Could it be that one moving observer could see an infinite number of computations occurring even though only a finite number had occurred according to someone else?
The famous motivating example of this sort is the so-called twin paradox. Two identical twins are given different future careers. Tweedlehome stays at home while Tweedleaway goes away on a space flight at a speed approaching that of light. When they are eventually reunited, relativity predicts that Tweedleaway will find Tweedlehome to be much older. The twins have experienced different careers in space and time because of the acceleration and deceleration that Tweedleaway underwent on his round trip. So can we ever send a computer on a journey so extreme that it could accomplish an infinite number of operations by the time it returns to its stay-at-home owner?
Itamar Pitowsky, a philosopher of science at the Hebrew University of Jerusalem, argued that if Tweedleaway could accelerate his spaceship sufficiently strongly, then he could record a finite amount of the universe's history on his own clock while his twin records an infinite amount of time. Does this, he wondered, permit the existence of a "Platonist computer" - one that could carry out an infinite number of operations along some trajectory through space and time and print out answers that we could see back home. Alas, there is a problem - for the receiver to stay in contact with the computer, he also has to accelerate dramatically to maintain the flow of information. Eventually, the gravitational forces become stupendous and he is torn apart.
Notwithstanding these problems, a checklist of properties has been compiled for universes that can allow an infinite number of tasks to be completed in finite time, or "supertasks" as they have become known. These are called Malament-Hogarth (MH) universes after David Malament, the University of Chicago philosopher, and Mark Hogarth, a former Cambridge University research student, who in 1992 investigated the conditions under which they were theoretically possible. Supertasks open the fascinating prospect of finding or creating conditions under which an infinite number of things can be seen to be accomplished in a finite time. This has all sorts of consequences for computer science and mathematics because it would remove the distinction between computable and uncomputable operations.
It is something of a surprise that MH universes are possibilities but, unfortunately, have properties that suggest they are not realistic unless we embrace some disturbing notions, such as the prospect of things happening without causes and travel through time. The most serious by-product of being allowed to build an infinity machine is rather more alarming though. Observers who stray into bad parts of these universes will find that being able to perform an infinite number of computations in a finite time means that any amount of radiation, no matter how small, gets compressed to zero wavelength and amplified to infinite energy along the infinite computational trail. Thus any attempt to transmit the output from an infinite number of computations will zap the receiver and destroy him.
So far, these dire problems seem to rule out the practicality of engineering a relativistic infinity machine in such a way that we could safely receive and store the information. But the universes in which infinite tasks are possible in finite time include a type of space that plays a key role in the structure of the very superstring theories that looked so appealingly finite. Perhaps infinity still lurks in the wings ready to play a new and unexpected role in the drama of the universe.
John D. Barrow is professor of mathematical sciences at Cambridge University, Gresham professor of astronomy and author of The Infinite Book and The Book of Nothing , Jonathan Cape.